Oscillation of Even-Order Neutral Delay Differential Equations
© Tongxing Li et al. 2010
Received: 28 November 2009
Accepted: 29 March 2010
Published: 19 May 2010
By using Riccati transformation technique, we will establish some new oscillation criteria for the even order neutral delay differential equations , , where is even, , , and . These oscillation criteria, at least in some sense, complement and improve those of Zafer (1998) and Zhang et al. (2010). An example is considered to illustrate the main results.
In what follows we assume that
Neutral differential equations find numerous applications in natural science and technology. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines; see Hale .
Motivated by Liu and Bai , we will further the investigation and offer some more general new oscillation criteria for (1.1), by employing a class of function and operator The method used in this paper is different from .
2. Main Results
In this section, we give some new oscillation criteria for (1.1). In order to prove our theorems we will need the following lemmas.
Lemma 2.1 (see ).
Lemma 2.2 (see ).
Lemma 2.3 (see ).
which contradicts (2.3). This completes the proof.
We get the following new result.
By Theorem 2.4 (or Theorem 2.5) we can obtain the oscillation criterion for (1.1) (or (2.20)); the details are left to the reader.
For an application, we give the following example to illustrate the main results.
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (60774004, 60904024), China Postdoctoral Science Foundation funded Project (20080441126, 200902564), and Shandong Postdoctoral funded project (200802018) by the Natural Science Foundation of Shandong (Y2008A28, ZR2009AL003), and also by University of Jinan Research Funds for Doctors (B0621, XBS0843).
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